document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192817>Question 1192817</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A researcher wishes to estimate, with 90% confidence, the proportion of adults who have high speed internet access. Her estimate must be accurate within 8% of the true proportion. Find the minimum sample size needed, using a prior study that found that 67% of the respondents said they have high speed internet access.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824739 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824739\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "At 90% confidence, the z critical value is roughly z = 1.645<BR>\n" );
document.write( "Use a Z table to determine this.<BR>\n" );
document.write( "Notice that P(-1.645 < Z < 1.645) = 0.90 approximately.\r<BR>\n" );
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document.write( "The prior study says phat = 0.67 is the proportion of people who have high speed internet access (i.e. the 67%)<BR>\n" );
document.write( "E = 0.08 due to the desired 8% error. The researcher wants to achieve this error or smaller.\r<BR>\n" );
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document.write( "The min sample size needed is...<BR>\n" );
document.write( "n = phat*(1-phat)*(z/E)^2<BR>\n" );
document.write( "n = 0.67*(1-0.67)*(1.645/0.08)^2<BR>\n" );
document.write( "n = 93.4847  approximately<BR>\n" );
document.write( "n = <font color=red>94</font>  always round UP to the nearest whole number\r<BR>\n" );
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document.write( "Let's say we didn't round up and instead went with n = 93 (it's tempting because 93.4847 is closer to 93 than it is to 94)<BR>\n" );
document.write( "The margin of error would be roughly<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 1.645*sqrt(0.67*(1-0.67)/93)<BR>\n" );
document.write( "E = 0.0802<BR>\n" );
document.write( "We get an error larger than 8% which isn't what your teacher wants.\r<BR>\n" );
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document.write( "However, if we go with n = <font color=red>94</font>, then,<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 1.645*sqrt(0.67*(1-0.67)/<font color=red>94</font>)<BR>\n" );
document.write( "E = 0.0798<BR>\n" );
document.write( "Now the error is under 8%<BR>\n" );
document.write( "Likely we won't land *exactly* on 8%, so the next best thing is to get under that value.<BR>\n" );
document.write( "The larger n gets, the smaller E will get, and vice versa.\r<BR>\n" );
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document.write( "This is why we always round up for minimum sample size problems.\r<BR>\n" );
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document.write( "Answer: <font color=red>94</font><BR>\n" );
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